Gaussian Process Modeling of Protein Turnover - Journal of Proteome

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Gaussian Process Modeling of Protein Turnover Mahbubur Rahman, Stephen F Previs, Takhar Kasumov, and Rovshan G. Sadygov J. Proteome Res., Just Accepted Manuscript • DOI: 10.1021/acs.jproteome.5b00990 • Publication Date (Web): 27 May 2016 Downloaded from http://pubs.acs.org on May 30, 2016

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Journal of Proteome Research

Gaussian Process Modeling of Protein Turnover Mahbubur Rahman1,2, Stephen F. Previs3, Takhar Kasumov4,5, and Rovshan G. Sadygov1,2,* 1

Department of Biochemistry and Molecular Biology, 2Sealy Center for Molecular

Medicine, The University of Texas Medical Branch, Galveston, Texas 77555 3

Merck Research Laboratories 2015 Galloping Hill Road Kenilworth, NJ 07033

4

Department of Gastroenterology & Hepatology, Cleveland Clinic Cleveland, OH 44195 5

Department of Pharmaceutical Sciences

School of Pharmacy, Northeast Ohio Medical University Rootstown, OH 44225 *To whom correspondence should be addressed: Tel. 409-772-3287; Fax 409772-9670; email: [email protected] Keywords: dynamic proteome, protein degradation rate constant, protein turnover rate constant, Ornstein-Uhlenbeck process, Gaussian Process, stochastic differential equation for protein turnover rate constant, mass spectrometry, stable isotope labeling.

Abbreviations: GP, Gaussian Process; SSM, subsarcolemmal

mitochondria; LC-MS, liquid-

chromatography-mass spectrometry; ODE, ordinary differential equation; OU, OrnsteinUhlenbeck.

1 ACS Paragon Plus Environment


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Abstract

We describe a stochastic model to compute in vivo protein turnover rate constants from stable isotope labeling and high-throughput liquid-chromatography-mass spectrometry experiments. We show that the often used one- and two-compartment non-stochastic models allow explicit solutions from the corresponding stochastic differential equations. The resulting stochastic process is a Gaussian Processes with Ornstein-Uhlenbeck covariance matrix. We applied the stochastic model to a large scale data set from

15

N labeling and compared its performance metrics with that of the non-

stochastic curve fitting. The comparison showed that for more than 99% of proteins the stochastic model produced better fits to the experimental data (based on residual sum of squares). The model was used for extracting protein decay rate constants from mouse brain (slow turnover) and liver (fast turnover) samples. We found that the most affected (compared to two-exponent curve fitting) results were those for liver proteins. The ratio of the median of degradation rate constants of liver proteins to those of brain proteins increased by four-fold in stochastic modeling compared to the two-exponent fitting. Stochastic modeling predicted stronger differences of protein turnover processes between mouse liver and brain than previously estimated. The model is independent of the labeling isotope. To show this we also applied the model to a protein turnover studied in induced heart failure in rats where metabolic labeling was achieved by administering heavy water. No changes in the model were necessary for adapting to heavy water labeling. The approach has been implemented in a freely available R code.


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Introduction

The continuous degradation and synthesis of proteins plays an important role in maintaining cellular homeostasis1. The dynamic equilibrium of cellular protein abundances can change due to, for example, external stimuli, developmental programs, onset of diseases, or ageing. The new equilibrium is achieved by balancing between alterations in the rate of synthesis and/or the rate of degradation. The combined process of synthesis and degradation termed as protein turnover refers to the movement of a protein through its pool and is often expressed as a first-order rate constant2. It is thought that protein degradation is a stochastic process – all proteins (newly synthesized and old) have the same probability of being degraded2. Labeling of proteins with radioactive amino acids is considered to be a “golden standard” for studying proteome dynamics. In this, “pulse-chase” method, proteins of interest are radioactively labeled over a brief period of time (the pulse), and then the decay in radioactivity is measured over time (the chase) and is used to determine protein half-life3. This assay is accurate and minimally perturbative but requires a specific antibody for each protein which makes it difficult for scaling to multiple proteins. Fluorophore tagging of proteins have also been used for proteome dynamics studies4,5. While these methods are real-time, it has been argued that the tagging may affect protein turnover properties, both through changing the protein structure and by altering their abundance and stoichiometry relative to interaction partners6. Mass spectrometry based proteomics is a widely used, high-throughput technology to identify and quantify proteins7. Most of the quantitative proteomics



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applications so far have been used for measuring the net change in protein abundance8. The technology has recently been enhanced and applied9 in studies aimed at

measuring the rate of protein turnover both in vivo10-15 and in vitro16-19. Protein

turnover

studies

using

metabolic

labeling

and

Liquid-Chromatography

Mass

Spectrometry (LC-MS) analyses have a complex design that includes a critical bioinformatics component to extract protein decay rate constants2. The work discussed herein focuses on the application of a stochastic model to extract protein turnover rate constants from in vivo protein turnover studies. Briefly, the metabolic incorporation of stable isotope and LC-MS measurements of time-course peptide/protein labeling provide data on protein kinetics. Traditionally, exponential decay curves have been used to fit the experimental data and infer the protein turnover rate constants11,18,20,21. Single- (single exponential) and multi-compartment (first order synthesis, and multi-exponential21,22) models have been employed. Time-dependent ordinary differential equations (ODE) have also been used to model the stable isotope incorporation in metabolic labeling22,23. However, the general applicability of these models to in vivo studies are limited, as they may result in negative decay rates (for multi-compartment systems) and are not able to explain fluctuations in the experimental data21. Stochastic models were recently applied to high-throughput time-course data in proteome studies24,25. SORAD24 used a Gaussian Process (GP) to describe time-course data of phosphor-proteins from enzyme-linked immunosorbent assay experiments. Similar models have been used for chromatographic alignment25, and beta-binomial process modeling of high throughput sequencing26. The stochastic processes add the


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flexibility of accounting for fluctuations in the data. Here we describe a non-parametric, data-driven stochastic approach - GP for modeling time-course data from metabolic labeling experiments. We show that the GP with the Ornstein-Uhlenbeck (OU) kernel is a natural solution to the protein turnover kinetics in one- or two-compartment models. This approach produces fits that are highly improved as compared to the traditional curve fitting models, it overcomes problems with negative decay rates and random fluctuations in decay rates. Last, the model is isotope independent (i.e., labeling).

The

R

code

implementing

the

model

is

15

N or 2H

available

at

https://ispace.utmb.edu/users/rgsadygo/Proteomics/ProteinKinetics.

Methods Protein turnover studies using metabolic labeling, mass spectrometry and bioinformatics have a complex design and comprise multiple stages27. There are some variations dependent on what type of labeling (e.g.

15

N, 2H,

13

C) is used and how a

tracer is administered. For studies that rely on heavy water labeling, for example, the animals are provided heavy water using a “primed-infusion” logic. For example, animals are given an initial loading bolus in order to instantaneously raise the enrichment of body water to a desired level. They are simultaneously given labeled water in the drinking bottles to maintain the enrichment. At pre-determined time points, the animals are sacrificed and the organs or tissues are collected. LC-MS/MS is used to identify proteins and measure the level of incorporation of heavy isotopes28-30. The time-course data of the heavy isotope incorporations is then modeled to extract the protein turnover rate constant, Figure 1. The last stage is the focus of this study. In the following text, we



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will first review the current models and then describe our stochastic model to extract protein turnover rate constants from heavy isotope labeling data. Ordinary differential equations have previously been used to model the incorporation of the heavy isotope labeled species in metabolic labeling and LC-MS21,23. As a prototypical study, Guan and colleagues21 have used one- and two-compartment models to determine protein turnover rate constants from experiments using 15N-labeled algae diet (produced from

15

N-enriched NaNO3) and mass spectrometry. In brief, the

one-compartment model assumes that proteins are synthesized at a constant rate ksyn and degraded proportional to their abundance levels with the degradation rate constant of kdeg. With the additional assumption of the steady-state for a protein concentration, it was shown that the fraction of the heavy isotope labeled proteins, β(t) (characterized by the relative isotope fraction, RIF), is described by a first order deterministic equation: = k − k βt βt (1) where t is the time, and the initial value condition is: β(0) = 0(the baseline, before the tracer administration). The solution of this equation is a growth curve with a single exponent: βt = 1 − e The details of all formula derivations are provided in the Supplementary Information. We note that under the steady-state assumption (which is used in this work), protein degradation rate constant is equal to the fractional protein synthesis rate (ratio of the synthesis rate over the protein concentration). Thus, throughout the paper, we refer to the protein turnover rate constant and degradation rate constant interchangeably.


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Several other publications11,12,18,20,31 modeling protein turnover from stable isotope labeling experiments have used equations similar to Eq. (1). However, when applied to the data sets from mouse protein turnover studies that used

15

N labeling, the fit to the

data was often complicated (in particular for fast protein turnover)21. Better results were achieved by using a two-compartment model. In this model, the first compartment considers the kinetics of amino acids, while the second compartment accounts for the amino acids incorporation into proteins. Protein synthesis is assumed to be a first order process. It is modeled to depend on the concentrations of heavy isotope labeled amino acids (precursor pool). The model leads to a system of equations for the fraction of heavy isotope labeled amino acids, α(t), and β(t):

= k − k αt αt = k αt − k βt βt

(2) where ksyn is the rate constant of protein synthesis from the precursor amino acids. Eq. (2) was derived under the assumption of the steady-state for total (labeled and unlabeled) protein level. The amino acids are synthesized with the constant rate, k0, and degraded proportional to their abundances with the rate constant, ksyn. Protein degradation is characterized with the rate constant, kdeg. The equation for α(t) is solved first, with the initial value for stable isotope labeled amino acids, α(0) set equal to zero: αt = 1 − e which leads to the following equation for the β(t): = k 1 − e − k βt βt (3)



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The solution of Eq. (3) (β(0) = 0) is the two-exponent growth curve: βt = 1 − k e − k e /k − k The solutions for α(t) and β(t) were obtained with the assumption that the heavy isotope labeling is asymptotically complete. This assumption, while somewhat reasonable for incorporation of

15

N-labeled amino acids into proteins, will not be generalizable for

incorporation of deuteriums during administration of heavy water. This is because, in contrast to fully

15

N-labeled diet, the metabolic labeling with heavy water involves

administration of only ~5% 2H2O in the drinking water. Metabolic labeling with 100% 15N in the diet may lead to the complete labeling of all nitrogen atoms in a protein, but using 5% heavy water will only result in partial labeling of the proteins. The incomplete labeling was also observed6 in labeling by SILAC32. It was observed that unless the cell medium is regularly renewed by labeled amino acids, complete labeling of proteins will not be achieved. In animal experiments it is not feasible to renew the cellular environment. To account for the incomplete labeling, we introduced a new parameter, heavy isotope enrichment fraction at the asymptote, λ. With this parameter, the solution for β(t) becomes: βt = λ − λk e − k e /k − k (4) A more descriptive text on the one- and two-compartment models and their solutions is provided in the Supporting Information section. Even though, the two-compartment model produced significantly improved fits to the experimental data, it was observed that the model was not able to account for the stochastic fluctuations in the


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experimental data21. It has subsequently lead to high synthesis rate constants in the two-compartment model and/or negative decay rate constants (for the expanded, threecompartment model). Here, we will show that stochastic differential equations based on either Eqs. (1) or (2) allow exact solution in the form of a GP with OU kernel. The stochastic equation uses a model fluctuation parameter that allows one to account for and estimate the effects of stochastic fluctuations in experimental data. The stochastic analog of Eq. (3) for β(t) (with the inclusion of the incomplete isotope labeling at the asymptote) is: dXt = k λ1 − e − Xtdt + Γk , k , t, X, λ, σ' dBt; X0 = 0 (5) where X(t) is a stochastic variable denoting the fraction of the heavy isotope enriched protein (β(t) in the deterministic model), Γk , k , t, X, λ, σ' is the volatility which, in general is dependent, on the model parameters ksyn, kdeg, λ, X(t), and the variance of the model fluctuations, σγ; B(t) is the Brownian motion (White Gaussian noise). If we assume that in addition to the model fluctuations there are also measurement errors that are described by a White Gaussian noise, ε (with standard deviation σε), the final model is: Yt = Xt + ε; ε~N0, σ0/ . The model depends on the stochastic variable, X(t), which is the solution to Eq. (5). We assume a homoscedastic variance which allows for an explicit solution to Eq. (5). Thus if the volatility, Γ, depends only on the model fluctuations, σγ, the Eq. (5) becomes: dXt = k λ1 − e − Xtdt + σ' dBt (6) 9 ACS Paragon Plus Environment


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Eq. (6) can be solved (using the assumption of the Itô’s process33,34 for X(t) and the transformation g(X,t) = X(t)*exp(kdegt). The details of the derivation are provided in Supplementary Information section). The solution of Eq. (6) is:

Xt = λ − λk e − k e /k − k + σ' 23 e dBs (7) Note that the solution, (7), is the OU process34,35 with a mean value, β(t), in Eq. (4), and covariance matrix, K(t,s) (also called Gram matrix), expressed as: Ks, t = σ0' 6 || /k

where t and s are time points. The result allows direct application of the OU process to the proteome turnover dynamics. Since the solution is exact, it is not only methodologically preferable but also offers practical advantages to the Gaussian kernel model that was previously used for the first order equation24, this stems from the fact that for the exact solution there is one less parameter. The scaling factor in the Gaussian kernel is an independent parameter that needs to be estimated from the data. In the exact solution, the scaling parameter in OU kernel is the degradation rate constant, kdeg. In addition, it is natural that for the processes that behave exponentially, Eq. (7), the covariance terms between the time points will also be exponential (depending on the absolute value of the time difference), rather than Gaussian. Thus, our final model for the heavy isotope labeled protein proportions becomes: 89, μ 8Y9~MVNμ 89, Σ; Y 89 ∈ R ; 8 where MVN stands for multivariate normal distribution, n is the number of data points (time points at which heavy isotope levels have been measured), µ is equal to β which is determined in (4), and the covariance matrix, ∑ is an n by n matrix defined as: 10 ACS Paragon Plus Environment

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∑s, t = Ks, t + σ0/ δs, t (9) where δ is the Kronecker’s delta. The model (8) with the covariance matrix (9) is a GP with OU kernel and numerical methods exist36,37 for its solution to obtain kdeg and ksyn. The approaches to extract the parameters from the data use the posterior distributions or likelihood function and find parameter values that maximize them. In total there are five hyperparameters in our model, θ = k , k , σ' , σ/ , λ. The log-likelihood function is: logℒθ = −0.5 logdetΣ − 0.5y 89 − μ 89J Σ K y 89 − μ 89 − 0.5nlog2π

(10)

The posterior distribution, pθ|y, of the parameters is proportional to the product of the

likelihood function and the prior distributions, πθ. The log of the posterior distribution of the hyperparameters is: logpθ|y ∝ logℒθ + log πθ The hyperparameters can be estimated from the data either by

maximizing the

posterior distribution with respect to the hyperparameters36, or by sampling from the posterior distribution with Markov chain Monte Carlo methods25. We used a quasiNewton approach, Broyden–Fletcher–Goldfarb–Shanno algorithm38 available in R39, to maximize the posterior distribution function and obtain the hyperparameters. For the prior distributions (except for λ) we chose exponential distributions with the prior decay rates estimated from the non-stochastic model. A uniform prior was chosen for the asymptotic enrichment level, λ. Note that the derivative of the log-likelihood function, Eq. (10), with respect to the every hyperparameter is in a closed form and is used in the



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optimizations. The corresponding formulas are shown in the Supplementary Information section. The approach is implemented in an R39 code and is available in the Supplementary Materials.

Results Performance metrics using 15N labeling data. The protein turnover experiments using

15

N labeling were used to test and compare

the performance of the stochastic model with the results from exponential curve fitting. The 15N labeling data is freely available on-line21. The data consists of two parts, a brain and a liver proteome. For every protein, RIF values of all peptides (for the protein) have been averaged to obtain protein RIF. We used these values to test our model. On average, protein turnover rates in brain are longer than those in the liver. For the nonstochastic model, fitting of liver data has been more difficult and required three exponential curves, some of which produced negative rate constants21. As a goodness of fit metric we chose the residual sum of squares (RSS) for each model: QRR = S

yT − yUT 0

TVK

where WX are the observed values and WUX are the corresponding theoretical predictions. In Figure 2, we show the comparison of the RSS values between the stochastic modeling (x axis) and the two-exponent curve fitting (y-axis) for mouse liver proteome (the red line is the line of unity). For the data points above the line of unity, RSSs of the two-exponent curves are larger than the corresponding RSSs from GP. The fits obtained by the GPs resulted in smaller RSS’s for 794 out of 797 proteins. Unlike the 12 ACS Paragon Plus Environment

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exponential curve fitting, GP allows for correlations between the measured RIFs in the time-course data. It is a desirable feature as the data points that are nearby in the time domain are expected to be correlated and ideally a model will be able to account for the correlation. In the case of the GP, the time correlations are provided in the structure of the Gram matrix. The diagonal elements of the matrix are the sum of the white noise variance and amplitude of the OU kernel. The non-diagonal elements of the matrix are the covariance between the different time points. The corresponding figure for brain proteome is shown in the Supplementary Materials Figure S1. Another goodness-of-fit measure in time course models is the Pearson correlation between the predicted values and experimental observations along the time axis. In Figures S2 (mouse liver) and S3 (mouse brain) of the Supplementary Materials Section, we show the scatter plot of the Pearson correlation coefficients for each protein obtained using predictions from twoexponent fitting and GP modeling. Similar to RSS, GP showed substantially improved correlation coefficients compared to the two-exponent fitting. To explicitly compare the fits from the two models we have chosen a liver mitochondrial protein, trifunctional protein subunit α (Q8BMS1). The experimental data (empty circles) and fits (green line – GP, blue line – exponential curves) are shown in Figure 3. The GP results in a better fit (approximately, 70 times smaller RSS value compared to the two-compartment exponential curve fit). The ksyn rate constant computed in the exponential curve fit is very large, in the order of 1012 (day-1), while the GP produced 1.4 day-1. The degradation rate constant, kdeg, in GP was 0.15 day-1 which was twice higher than the corresponding value from the two-exponent curve fit. The 95% confidence interval of the GP predicted fit is shown in Figure S4.



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Comparison with the ODE results using 15N labeling data. The scatter plot of the computed decay rate constants using a GP model and twoexponent curve fitting for mouse liver proteins is shown in Figure 4. The relevant data for mouse brain are shown in Supplementary Materials Figure S5. In general, the GP model produced decay rate constants that are larger than those from the two-exponent curve fitting. The correlation between the rate constants was 0.81. The density plot of the differences between kdeg values produced by the GP and two-exponent curve fit are shown in Supplementary Figure S6. The distribution peaks at about 0.18 day-1 and extends up to 1 day-1. For liver proteins, the median degradation rate constants were 0. 091 day-1 and 0.228 day-1 for two-exponent curve fit and GP, respectively. Thus, GP modeling, on average, predicts more than two fold faster degradation rate constants for liver proteins. For the mouse brain proteins the effects were not so drastic. The median rate constants for brain proteins were 0.040 day-1 and 0.054 day-1 in two-exponent curve fit and GP modeling, respectively. It has been noted previously that the exponent curve fitting models were less accurate for modeling faster protein turnover (liver proteins)21. The ratio of the median turnover rate constants of liver proteins to brain proteins increased from 2.25 in two-exponent fitting to 4.23 in GP modeling. Thus the GP modeling predicts a larger difference between the protein turnovers in liver and brain. The ratio of the medians of turnover rate constants predicted by the GP modeling compares well to earlier studies that estimated the rate of incorporation of [14C]tyrosine in mouse liver (2.8% per hour) and brain (0.6% per hour)40. Note that in these studies the calculation of the rate constants are based on the tracer incorporation rate and the steady-state assumption, i.e. the fractional synthesis rate of proteins are equal to the fractional degradation rate. Under the steady-state assumption, the degradation rate 14 ACS Paragon Plus Environment

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constant is linearly dependent on the synthesis rate as can be seen from the onecompartment model (solution of Eq. (1b) in Supplementary materials). Figure S7 summarizes decay rate constants of brain and liver proteomes computed using twoexponent curve fitting and GP modeling. Stronger differences were observed in cases where one estimates the synthesis rate constants, ksyn, Figure 5. Unlike the two-exponent curve fit, there are no ksyn values by the GP that are larger than 10 day-1. For the two-exponent curve fitting ksyn values as large as 1014 day-1 have been produced. The median values of ksyn for two-exponent curve and GP were 9.12*109 day-1 and 1 day-1, respectively. We note that in the labeling experiments where the first isotope distribution pattern is measured after about nine hours of metabolic labeling, accurate determination of the rate constants of this much higher magnitude are unrealistic. The GP on the other hand yielded a maximum synthesis rate constant less than 10 day-1. In addition, as it is shown in Figures 2 and S1-S3, the goodness of fit is substantially improved for the GP. The synthesis and degradation rate constants as computed by GP and two-exponent curve fitting are provided in supplementary materials Tables S1 and S2 for liver and brain proteins, respectively. The stochastic model produces a volatility standard deviation, σγ, for each protein rate constant estimate. The higher volatility standard deviation means a stronger random fluctuation effects. In Figure S8 of Supplementary Materials we show the distribution of model standard deviations for the liver proteins. The mode of the standard deviation is non-zero at 0.018, the highest values are at 0.08.



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As it was mentioned above one way to improve the model fitting to the data has been to add more compartments. However, we believe that one should consider the development of a rationale solution and interpretation, e.g. make sure the modeling is done properly (as Guan et al21 have shown) but that it is supported by some strong physiological underpinnings. It would appear that the addition of more compartments solved one problem (i.e. gave a better fit of the data) but failed to address what these compartments would represent from a biochemical perspective. Our approach simply considers another way to align the biology and the modeling. We suspect that the approach described herein provides a better fit than previous ODE methods, in part, because of instability in the 15N-precursor precursor labeling (note that a major assumption is a constant precursor labeling). For example, if the amino acid enrichment is constant one would expect that the use of an ODE method should produce a good fit of the apparent protein decay rate constant. Presumably the use of 15

N-labeling (via the diet) results in pulsatile (or oscillatory) changes in

proteogenic amino acids. One would expect a maximum

15

N-labeling of

15

N-enrichment immediately

following consumption of the food and a lower degree of labeling at times when food is not consumed (since during those periods

14

N would be appearing via endogenous

protein degradation). In addition, one needs to consider that there are gradients between plasma amino acid labeling and the labeling of free amino acids in different tissues. Effectively, the “free amino acid” compartment is further compartmentalized and the “mixing” between different tissues and variable plasma labeling is not necessarily equal across sites. We suspect that this is what led Guan and colleagues21 to use models of increasing complexity when fitting the tissue-specific protein labeling. The


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observations would be expected to play a greater role in cases where protein turnover is faster (e.g. liver vs brain) and therein require more complex models (e.g. three- vs twocompartment) to get a better fit.

Kinetics of heart mitochondrial Carnitine Palmitoyltransferase 1 In a recent study41 we used heavy water labeling to determine the changes in halflives of mitochondrial proteins in cardiomyocytes due to the induced heart failure in rats. One of the proteins that we studied was carnitine palmitoyltransferase (CPT) 1. The experimental data (crosses – normal heart, triangles – induced heart failure) and corresponding GP fits (black line normal heart, blue line – induced heart failure) are shown in Figure 6 (the data is for the CPT 1 from subsarcolemmal mitochondria (SSM) of cardiomyocytes). The computed turnover rate constants for the normal and induced heart failure samples were practically the same and equal to 0.09 day-1. The result is in agreement with the finding that turnover rates for CPT 1 in SSM were not affected by the induced heart failure. The ksyn rate constant was substantially larger in heart failure induced SSM CPT 1 than in the protein of the normal heart: 2.2 day-1 vs. 0.09 day-1.

Conclusions and Future Work We have applied a GP modeling to extract protein turnover rates from time-course data of isotope incorporation in stable isotope labeling and LC-MS experiments. It was shown that an exact solution to the stochastic differential equation yields a GP solution with OU kernel. The approach is preferable to the more widely used GP with Gaussian kernel, as the OU solution is exact, and the number of parameters is less. We then used



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this approach to analyze a large scale data set obtained from mice fed a diet containing 15

N-labeled algae. The results have demonstrated that GP modeling substantially

improves the fit to the experimental data, particularly for protein turnover in liver, which in a previous model produced negative rate constants. We also applied the model to carnitine palmitoyltransferase of SSM of cardiomyocytes that were studied to infer effects of the induced heart failure. This experiment had used heavy water labeling. In future studies we plan to extend the model to include relative isotope fractions for each peptide of a protein. A similar approach has employed a mixed-effects modeling for growth curves resulting from the Gompertz model42. Using each peptide data separately rather than averaging peptide data will allow to estimate the variability due to the heavy isotope labeling of peptides. In addition, we will extend the model to describe the turnover of proteins that are not in a steady-state condition43.

Acknowledgements

Research reported in this publication was supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number R01GM112044.

Supporting Information Available: Supplementary materials include “Derivation of the GP Model from the Stochastic Differential Equation of Protein Turnover Kinetics”; Diagram D1 − one- and two-compartment models of protein turnover; Figure S1 − pairwise scatter plot of the RSS for mouse brain proteins; Figures S2 and S3 − the


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pairwise scatter plots of the Pearson correlations obtained from GP and two-exponent curve fitting for mouse liver and brain proteins, respectively; Figure S4 − 95% confidence interval of the mean of GP model for mitochondrial trifunctional protein, subunit α (Q8BMS1); Figure S5 − the scatter plot of the decay rates constants obtained from the GP and two-exponent curve fitting for mouse brain proteins; Figure S6 − the density of the difference between degradation rate constants computed by the GP and two-exponent curve fitting; Figure S7 – the boxplots of decay rate constants computed by two-exponent curve fitting and GP model for mouse brain and liver proteomes; Figure S8 –the density of standard deviations of the model distributions for mouse liver proteins; and Tables S1 and S2 – synthesis and degradation rate constants as computed by GP and two-exponent curve fit for mouse liver and brain proteins, respectively.



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Figure. 1. Modeling of protein turnover rate constants from stable-isotope labeling experiments. The time-course of heavy isotope labeling, extracted from the full scan spectra of tryptic peptides, is modeled as a GP with OU kernel.


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RSS, Exponential Curves

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Figure 2. Comparison of the residual sum of squares obtained by using exponential curve fitting44 (y-axis) and GP (x-axis). The red line is the line of unity. The shown data is for the liver proteome.


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Figure 3. Experimental (empty circles) data and model fits (green line – GP, blue line – exponential curve fitting) for mitochondrial trifunctional protein, subunit α (Q8BMS1). The GP produced a fit with RSS about 70 times smaller than that from the two-exponent fit.


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Figure 4. A scatter plot of the decay rates constants as computed by the GP (x-axis) and two-exponent curve fitting (y-axis). In general the rate constants computed by the GP model were larger (mouse liver proteins).


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Figure 5. Synthesis rate constants as computed using two-exponent curves (y-axis) and GP (x-axis) for mouse liver data proteome21.


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Figure 6. A GP modeling of RIF for CPT protein from normal (experimental data is shown with blue triangles and the fit in blue line) and induced heart failure (the experimental data is shown with black crosses and the fit in black line) interfebrullar mitochondria41.



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Table of Contents:

Synopsis: We describe a stochastic modeling of protein turnover rate constants from stable isotope labeling and LC-MS. We show that the often used one- and twocompartment non-stochastic models allow an explicit solution from the corresponding stochastic differential equations. The resulting stochastic process is a Gaussian Process with Ornstein-Uhlenbeck covariance matrix. We applied the model to large scale data sets from

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N labeling and compared the performance metrics from the non-

stochastic curve fitting with the corresponding result from the appropriate stochastic model. The results showed that for more than 99% of proteins the stochastic model produces a better fit to the experimental data. The stochastic modeling predicted stronger differences in the protein turnover processes in the mouse liver and brain than previously estimated.



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Pearson Correlations (Two-Exponent Fit)

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The pairwise scatter plot of the Pearson correlations obtained from protein turnover models using exponential curve fitting (y-axis) and GP (x-axis). The red line is the line of unity.


Gaussian Process Modeling of Protein Turnover - Journal of Proteome

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